Square Root Calculator (2024)

Use this online calculator to easily calculate the square root of a given number, including fractions. Quick and easy square root finder.

Quick navigation:

  1. What is a square root?
  2. How calculate a square root
  3. Properties and practical application of square roots
  4. Commonly used square roots
  5. Does the calculator support fractions?

What is a square root?

The square root of a number answers the question "what number can I multiply by itself to get this number?". It is the reverse of the exponentiation operation with an exponent of 2, so if r2 = x, then we say that "r is the root of x". Finding the root of a number has a special notation called the radical symbol: √. Usually the radical spans over the entire equation for which the root is to be found. It is called a "square" root since multiplying a number by itself is called "squaring" as it is how one finds the area of a square.

For every positive number there are two square roots - one positive and one negative. For example, the square root of 4 is 2, but also -2, since -2 x -2 = 4. The negative root is always equal in value to the positive one, but opposite in sign. You can see examples in the table of common roots below. Most often when talking about "the root of" some number, people refer to the Principal Square Root which is always the positive root. This is the number our square root calculator outputs as well.

In geometrical terms, the square root function maps the area of a square onto its side length. The function √x is continuous for all nonnegative x and differentiable for all positive x.

How calculate a square root

Unlike other mathematical tasks, there is no formula or single best way to calculate the square root of a real number. Still, there is a handy way to find the square root of a number manually. The algorithm to find √a is as follows:

  1. Start with a guess (b). If a is between two perfect squares, a good guess would be a number between those squares.
  2. Divide a by b: e = a / b. If the estimate e is an integer, stop. Also stop if the estimate has achieved the desired level of decimal precision.
  3. Get a new guess b1 by averaging the result of step #2 e and the initial guess b: b1 = (e + b) / 2
  4. Go to step #2 using b1 in place of b

For example, to find the square root of 30 with a precision of three numbers after the decimal point:

Step 1: a = 30 is between 25 and 36, which have roots of 5 and 6 respectively. Let us start with b = 5.5.
Step 2: e = a / b = 30 / 5.5 = 5.45(45). Since b is not equal to e (5.500 ≠ 5.454), continue calculation.
Step 3: b1 = (5.45 + 5.5) / 2 = 5.47727(27)
Step 4: e = 30 / 5.47727 = 5.477178. Since b1 = e = 5.477 within three position after the decimal point, stop the square root-finding algorithm with a result of √30 = 5.47727(27).

Checking the outcome against the square root calculator output of 5.477226 reveals that the algorithm resulted in a correct solution. While the above process can be fairly tedious especially with larger roots, but will help you find the square root of any number with the desired decimal precision.

Properties and practical application of square roots

Square roots appear frequently in mathematics, geometry and physics. For example, many physical forces measured in quantities or intensities diminish inversely proportional to the square root of the distance. So, gravity between two objects 4 meters apart will be 1/√4 relative to their gravity at 0 meters. The same is true for radar energy waves, radio waves, light and magnetic radiation in general, and sound waves in gases. It is usually referred to as the "inverse-square law".

The square root is key in probability theory and statistics where it defines the fundamental concept of standard deviation.

Square Root Calculator (1)

Plotting the results from the square root function, as calculated using this square root calculator, on a graph reveals that it has the shape of half a parabola.


Commonly used square roots

Table of commonly encountered square roots:

Commonly used square roots
Square Root ofPositive rootNegative root
√11-1
√21.4142-1.4142
√31.7321-1.7321
√42-2
√52.2361-2.2361
√82.8284-2.8284
√93-3
√103.1623-3.1623
√164-4
√255-5
√366-6
√497-7
√648-8
√819-9
√10010-10
√102432-32

The calculations were performed using this calculator.

Does the calculator support fractions?

Yes, simply enter the fraction as a decimal number (use dot as a separator) and you will get the corresponding root. For example, to compute the square root of 1/4 simply enter 0.25 in the number field, press "Calculate" and you will get 0.50 as ouput. If you are having difficulty converting a fraction to a decimal number, you will find our fraction to decimal converter handy.

Square Root Calculator (2024)

FAQs

Is there any trick to find square root? ›

Trick to find Square Root of a 4-digit Number

Group the Digits: Group the digits of the 4-digit number into pairs from the left. Find the Nearest Perfect Square: Find the largest perfect square less than or equal to the first pair of digits. This will help you determine the first digit of the square root.

How many answers do square roots have? ›

Square root of a number always has two answers, one is positive and one is negative. There are two answers because the square of a negative and positive number is the same.

How to calculate square root manually? ›

Long division method
  1. Separate your square root base into pairs. ...
  2. Find the largest square that divides into the first number or pair. ...
  3. Subtract the square from the first number or pair. ...
  4. Drop down the next pair. ...
  5. Multiply the first digit of the square by two. ...
  6. Set up the next factor equation.
Jul 1, 2024

How to quickly estimate square root? ›

  1. By hand, the fastest way is to estimate the answer, the divide by it. Average that answer with the original guess, and you are close.
  2. Example: X = sqrt(70) = 8 (guess)
  3. Y = 70 / X = 70/8 = 8.75.
  4. Estimated answer = (8 + 8.75)/2 = 8.375.
  5. To get closer to the answer, repeat this process.
Apr 3, 2021

Is there a formula for square root? ›

Suppose x is the square root of y, then it is represented as x=√y, or we can express the same equation as x2 = y. Here, '√' is the radical symbol used to represent the root of numbers. The positive number, when multiplied by itself, represents the square of the number.

What is the easiest way to find the square root without a calculator? ›

To find the square root of a given square number by prime factorization, we follow the following steps:
  1. Obtain the prime factorization of the given natural number.
  2. Make pairs of identical factors.
  3. Take one factor from each pair and find their product. The product so obtained is the square root of the given number.

How to find square easy trick? ›

Step 1: Subtract the last digit from the number being squared. Step 2: Add the last digit to the number being squared. Step 3: Multiply the numbers from Step 1 and Step 2.

What is the square root guessing method? ›

The Babylonian square-root algorithm
  1. Make an initial guess. Guess any positive number x0.
  2. Improve the guess. Apply the formula x1 = (x0 + S / x0) / 2. The number x1 is a better approximation to sqrt(S).
  3. Iterate until convergence. Apply the formula xn+1 = (xn + S / xn) / 2 until the process converges.
May 16, 2016

Are there two answers to every square root? ›

Thus, any equation we solve by squaring the radical will provide us with two answers, one of which will always be extraneous.

Is a square root an exact answer? ›

The square root calculator finds the square root of the given radical expression. If a given number is a perfect square, you will get a final answer in exact form. If a given number is not a perfect square, you will get a final answer in exact form and decimal form.

How to find square root without square root button on calculator? ›

Long division method
  1. Separate your square root base into pairs. ...
  2. Find the largest square that divides into the first number or pair. ...
  3. Subtract the square from the first number or pair. ...
  4. Drop down the next pair. ...
  5. Multiply the first digit of the square by two. ...
  6. Set up the next factor equation.
Jul 1, 2024

How do you find the full answer on a TI-84? ›

Press ~ to see the entire expression. Tip: Press the cursor without pressing y to move the cursor along the line. An arrow appears to the left of an answer if it scrolls off the screen. Press ~ and | before you enter another expression to display the entire answer.

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